An operator \(T\)  is said to \(\lambda \) -commute with \(A\) provided \(AT=\lambda TA\) , where \(\lambda \in \mathbb {C}\) . If \(T\ne 0\) , then \(T\)  is called an extended \(\lambda \) -eigenoperator of \(A\)  with extended eigenvalue \(\lambda \) .
Extended \(\lambda \) -eigenoperators appear in the proof of an extension of Lomonosov's famous result, which was obtained independently by Scott Brown and Kim, Moore and Pearcy [1]}, [2]}. They proved that if an operator has a compact extended \(\lambda \) -eigenoperator then it has a non-trivial hyperinvariant subspace.
The determination of the extended \(\lambda \) -eigenoperators plays an important role in the study of an operator, and some properties, like hypercyclicity, are transferred in some way to the commutant (see [3]}).
